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- Quantum States and Quantum Gates
- Multiple Qubits and Entangled States
- Quantum Gate Arrays
- Quantum Parallelism
- Examples of Quantum Algorithms
- Grover’s Unstructured Search Algorithm
- Schor’s Factoring Algorithm
- Quantum Error Correcting Codes

6. Quantum Computer Architecture

7. Quantum Computers and Complexity

0 1

|1> =

|0> =

Quantum States

For quantum computing we need only deal with finite quantum systems, and it suffices to consider only finite dimensional complex vector spaces with inner product.

A quantum bit (qubit) is a unit vector in a one-dimensional complex vector space.

We will use Bra/Ket notation (invented by Dirac) to represent these unit vectors.

- 0 1

0 1

1 0

0 1

-1 0

- 0
- 0 -1

Single-Qubit State Transformations

Identity Transformation

I: |0> |0>

|1> |1>

Complement

X: |0> |1>

|1> |0>

Negative Complement

Y: |0> - |1>

|1> |0>

XY Product (phase transformation)

Z: |0> |0>

|1> - |1>

0 1 0 0

0 0 0 1

0 0 1 0

Reversible Gates

Controlled Not

CNOT: |00> |00>

|01> |01>

|10> |11>

|11> |10>

CNOT CNOT* = I

CNOT is unitary

Controlled-Controlled-Not (Toffoli Gate)

The third bit is negated only if the first two are both 1

CCNOT : |00x> |00x> where |x> = |0> or |x> = |1>

|01x> |01x>

|10x> |10x>

|11x> |11y> where |y> = X|x> complement of qubit

Controlled-Not and Toffoli gates are both reversible and the unitary transformations that they implement cannot be decomposed into a tensor product of single qubit transformations.

The Toffoli Gate can be used to construct a complete set of Boolean connectives.

Negation

Conjunction

T|1,1,x> = |1,1,x>

T|x,y,0> = |x,y,xy>

The single-qubit Hadamard transformation

H: |0> (1/2) (|0> + |1>)

|1> (1/ 2) (|0> - |1>)

When applied to n bits individually, H generates a superposition of all 2n possible states – which can represent numbers from 0 to 2n – 1.

(H H …. H) |00…00>

= (1/ 2n)((| 0> + |1>) (|0>+|1>) … (|0>+|1>)

= (1/ 2n) |x> where the summation is from x = 0 to x = 2n - 1

Let |0> and |1> be any two basis vector (states) in the complex plane. And let |x> be any (unit) vector (state) in the plane. Then:

|x> = a|0> + b|1>

<x| = |x>* = a*<0| + b*<1|

<x|x> = 1 = (a*<0| + b*<1|)(a|0> + b|1>)

= a*a <0|0> + a*b<0|1> + b*a<1|0> + b*b <1|1>

= a*a + b*b = 1

Since <0|0> = <1|1> = 1 and <0|1> = <1|0> = 0

Consider again the transformation X: |0> |1>

|1> |0>

We can represent X as

X = |0><1| + |1><0|

X|0> = (|0><1| + |1><0|)|0> = |0><1|0> +|1><0|0> = |1>

X|1> = (|0><1| + |1><0|)|1> = |0><1|1> +|1><0|1> = |0>

All unitary transformations can be expressed in Bra/Ket notation

Multiple Qubit states are formed from the tensor product of single Qubit states

|x> = a|0> + b|1>

|y> = c|0> + d|1>

|x> |y> = (a|0> + b|1>) (c|0> + d|1>)

= ac(|0> |0>) + ad (|0> |1>) +bc (|0> |0>) +(bd(|1> |1>)

= ac|00> + ad |01> + bc|10> + bd|11>

The tensor product of an m dimension vector and a k dimension vector is an mk dimension vector

Consider the two 2-qubit states:

|> = 1/2(|00> + |11>) and

|> = 1/2(|00> + |01>)

|> is composed by the tensor product |0> 1/2(|0> + |1>)

Measurement of the second bit will yield |0> or |1> with a probability of ½ for each result, independently of whether the first bit has been measured or not. Measurement of the first bit will always yield |0>.

|> cannot be decomposed into a tensor product of two single qubits

It is an entangled state.

The first measurement completely determines the outcome of the second.

Quantum states cannot be copied or cloned!

Proof

Assume a unitary transformation U such that U|a0> = |aa>

Let |a> and |b> be orthogonal states, and

U|a0> = |aa> and U|b0> = |bb>

Now consider |c> = 1/2(|a> + |b>)

By linearity U|c0> =1/2(U|a0> +U |b0>) = 1/2(|aa> + |bb>)

But if U is a cloning transformation

U|c0> = |cc>=1/2(|a> + |b>) 1/2(|a> + |b>)

= ½(|aa> + |ab> + |ba> + bb>)

Contradiction!!

1-bit full adder

|c>

|c>

|x>

|x>

|y>

|y>

|0>

|s>

|1>

|1>

|0>

|0>

|c’>

|0>

|0>

|1>

Let |c> = |1>, |x> = |0>, |y> = |1>

|s> = |0>, |c’> = |1>

Quantum Gate Arrays

It is possible to construct reversible quantum gates for any classical computable functionfwith m input andk output bits.

There exists a quantum gate array that implements the unitary transformation Uf : |x, y> |x, y f(x)>, where indicates bitwise xor.

|x>

|x>

|y>

|y f(x)>

Uf

Quantum Gate Arrays

The previous transformation Uf is reversible

Uf + = Uf Uf+Uf = Uf Uf = I

|x>

|x>

|x>

|y f(x)>

|y f(x) f(x)>

|y>

But |y f(x) f(x)> = |y>

Superposition of Quantum States

Consider the Tofolli Gate

|x>

|x>

|y>

|y>

|0>

|x> |y>

Apply T, the Tofolli transform, to the superposition of all inputs.

T(H|0> H|0> |0> ) = ½(|000> + |010> + |100> + |111>)

Quantum parallelism – Applying the Tofolli transform to a superposition of all of the input states produces a superposition of all of the states in the “truth table”

Superposition of Quantum States

BUT

Only one of the superposed states can be extracted by measurement

0

=

1

T(H|0> H|0> |0> ) = ½(|000> + |010> + |100> + |111>)

|000> + |010> + |100>

0

|111>

1

Measurement of the output projects the superposition onto the set of states consistent with the result

In order to take advantage of quantum parallelism one must:

1. Transform the state in such a way as to amplify the values of interest – so that they have a higher probability of being selected during measurement

Grover’s unstructured search algorithm

2. Find common properties of ALL the states of f(x)

Shor’s factoring algorithm

1/2n|x,P(x)>

i=0

Grover’s Unstructured Search

Given an unstructured list of size N and a proposition P, find an x in the list such that P(x) is true.

Let UP be the Quantum gate that implements the classical function P(x), and let n be such that 2n N.

UP : |x,0> |x,P(x)>

UP operating on a superposition of all input states yields:

If there is a single state where P(xi) = 1 (true), the probability of reading that state in a measurement is just 1/2n

We need to increase the probability of selecting this state.

1. Prepare a register containing the superposition of all values 0…2n-1

2. Compute P(x i) on this register.

3. Change the amplitude ai to -ai for xi such that P(xi) = 1

average

- Apply inversion about the average to increase the amplitude of xi with P(xi) = 1.

average

5. Repeat steps 2 through 4 /4(2n ) times.

6. Read the result

UP|,b> = (1/2n)UP ( |x, b> + |x, b>)

x X0

x X1

Grover’s Search Algorithm

Step 3. Changing the sign

Let UP be the unitary transformation that implements

UP |x, b> = |x, b P(x)>

Apply UP to the superposition |> = (1/2n) x=0..n-1|x> and

Choose b = (1/2)(|0> - |1>)

Let X0 = {x|P(x) = 0}, and X1 = {x|P(x) = 1} then

UP|,b> = (1/2n)UP ( |x, b> + |x, b>)

x X0

x X1

= (1/2n+1)UP ( |x, 0> + |x, 0> - |x, 1> - |x, 1>)

x X0

x X1

x X0

x X1

= (1/2n+1)( |x, 0 0> + |x, 0 1> - |x, 1 0> - |x, 1 1>)

x X0

x X1

x X0

x X1

= (1/2n+1)( |x, 0 > + |x, 1> - |x, 1 > - |x, 0>)

x X0

x X1

x X0

x X1

= (1/2n)( |x > - |x >) |b>

x X0

x X1

Grover’s Quantum Search Algorithm

The amplitude of the states in X1 have been inverted

N-1

- ai|xi> (2A – ai)|xi>

i = 0

i = 0

(2/N) – 1 2/N 2/N ……… 2/N

2/N (2/N)-1 2/N ………. 2/N

D =

………………………………………...

2/N 2/N 2/N ………… 2/N

Grover’s Quantum Search Algorithm

Step 4. Apply inversion about the average

Where A denotes the average of the ai’s

This inversion can be performed by the N x N matrix

- 0 -1 0 …. 0
- 0 …………… 0
- 0 ………...0 -1

R =

Grover’s Quantum Search Algorithm

The transformation D is unitary (D*D = I) and can be decomposed into O(n) = O(logN) quantum gates.

D = WRW where W is the Walsh-Hadamard transformation and

0 0 0 …….. 0

0 ……………… 0

0 ……………0 0

R’ =

2/N 2/N 2/N ……. 2/N

2/N 2/N 2/N ….…. 2/N

2/N ………………….. 2/N

2/N ……………2/N 2/N

WR’W =

Grover’s Quantum Search Algorithm

Consider R = R’ – I, where I is the identity matrix and R’ is

Then WRW = W(R’ – I)W, and it is easily shown that

And WR’W – I = D

Factor an integer M

Step 1 – choose arbitrarily an integer a

if a is not relatively prime to M we have found a factor

else apply the rest of the algorithm.

Step 2 – choose m such that M 2m <M2

Use quantum parallelism to compute f(x) = ax mod M

for all integers from 0 to 2m –1

The function is encoded in the quantum state

(1/2m) |x,f(x)> where x ranges from 0 to 2m -1

Step 2 -- details

Repeatedly square a (mod M) {classically} to get a2ifor i m

This requires O(m) squarings and multiplications of m-bit numbers (mod M)

This gives way to a gate array that uses O(m log(m) loglog (m))

gates to multiply two m-bit numbers (using Schönhage-Strassen)

The multiplication algorithm then becomes:

f(x) := 1

for i = 0 to m – 1

if (xi == 1) then //where xi is ith bit from right of x

f(x) := f(x) * a2i(mod M) //subroutine 2

endif

endfor

Step 2 – details (cont.)

Subroutine 2

let c = a2i (mod M), and b = poweri, then

result := 0

for j = 0 to m – 1

if (bi == 1) then

result := result + 2ic (mod M)

endif

endfor

Where 2ic is precomputed and built into the gate structure. The inverse of this operation is then used to erase b.

0 otherwise

C x g(x)|x,u> where g(x) =

and C is a complex scale factor

Schor’s Factoring Algorithm

Step 3 – Construct a state that has the same period as f

Measure the last lg M bits of (1/2m) |x,f(x)>

The measurement projects the state space onto the subspace that is compatible with the measured value>

The state after measurement is:

where

f(x) = ax (mod M)

Schor’s Factoring Algorithm

m bits

|x>

|x>

|f(x)>

|0>

log M qubits

Measure last log M bits

State |x, f(x)> is collapsed to g(x) |x,u> where

u is the measured state, and g(x) = 1 if f(x) = u, else 0

amplitude

x

The above graph displays the values of x consistent with the measurement on |f(x)>

If only we could measure the separation between adjacent values, our problem would be solved!

But alas, quantum mechanics only allows a single measurement!! We must find a way of determining the periodicity of these states from a single measurement of the first m bits of our register.

UQFT

Measurement of transformed states

Schor’s Factoring Algorithm

The Fourier transform of this superposition of states that give the measured u as a value of f(x) maps into states in a frequency domain.

g(x)|x>

G(c)|c>

|x>

Measurement of f(x)

Step 4 – Perform a quantum Fourier transform on the resulting state

UQFT : x g(x)|x,u> x G(c)|c,u>

When the period r of the function g(x) is a power of 2, the resulting Fourier transform is

j cj |j(2m/r),u>

where the amplitude is 0 except at multiples of 2m /r. When the period r does not divide 2m, the transform approximates the exact case and most of the amplitude is attached to integers close to 2m /r

- 0 1 0 0
- 0 0 1 0
- 0 0 0 eik-j

Let Sj,k =

Schor’s Factoring Algorithm

Step 4 – details

where k –j = /2k-j

The quantum Fourier transform is then given by

Hm-1Sm-2,m-1 Hm-2Sm-3,m-1Sm-3,m-2 Hm-3 …. H1S0,m-1S0,m-2 …S0,1H0

Where the matrices are applied in reverse order (from left to right) and the transformed state is the bit-reversed state of state |c>

The quantum Fourier transform requires m(m-1)/2 gates.

Step 5 – Extract the period

Measure the state, and call the result v

v = j(2m/r)

In general j and r will be relatively prime and the period is not a power of 2. A continued fraction expansion (on a classical computer) will produce an integer value q, that is likely to be the period.

Step 6 – Find a factor of M

If q is even, use the Euclidean algorithm to check whether (aq/2 + 1) or (aq/2 - 1) has a non-trivial factor with M

The reason why (aq/2 + 1) or (aq/2 - 1)is likely to have a non-trivial common factor with M is as follows:

If q is the period of f(x) = ax mod M then

aq = 1 mod M since

aq ax = ax mod M for all x.

If q is even we can write

(aq/2 + 1) (aq/2 - 1) = 0 mod M

thus, if neither of these factors is a multiple of M, either has a non-trivial factor with M. (found by using Euclid’s algorithm on a classical computer)

Step 7 – Repeat the algorithm, if necessary

Reasons for having to repeat the algorithm

- The value of v was not close enough to a multiple of 2m/r
- The period r and the multiplier j have a common factor – so that the denominator q was a factor of the period and not the period itself
- Step 6 yields M as M’s factor
- The period of f(x) = ax mod M is odd

Schor shows (claims) that this procedure, when applied to a random a (mod M) yields a non-trivial factor of M with probability at least 1 – 1/2k-1 where k is the number of distinct odd prime factors of M.

Example

Let M = 21

Step 1 – choose a = 11 which is relatively prime to M

Step 2 – let M2 2m < 2 M2 441 2m < 882 then m = 9

find ax mod M on the superposition of states x = 0,…,2m - 1

Step 3 – Measure the last lg M qubits of the state

(1/2m) x|x, ax (modM)>

C xg(x)|x, u>

Step 4 – Apply Fourier transform to collapsed state

UQFT : xg(x)|x, u> G(c)|c>

Step 5. Measure the transformed state

Assume measurement gives v = 427

Since v and 2m are relatively prime, the period r will

probably not divide 2m, and continued fraction expansion

of v/2m yields q = 6. (This is probably the period of f )

Step 6. Since q is even, aq/2 + 1 or aq/2 – 1 is likely to have a non-

trivial common factor with M. (On a classical computer)

a = 11, q/2 = 3, M = 21

(113 + 1) = 1332 (113 - 1) = 1330

gcd(1332, 21) = 7 and gcd(1330, 21) = 3

One problem in constructing quantum computers is the problem of decoherence – the interaction of qubits with the external environment causing them to transform in an unintended (and non-unitary) fashion.

Steane estimates the decoherence of any system likely to be built to be 107 times too large to run Shor’s algorithm on a 130 digit number.*

Characterization of Errors

| > (e1I + e2 X + e3 Y + e4 Z)| > = i ei Ei | >

The possible errors for each single qubit consist of no errors (I), bit- flip errors (X), phase errors (Z), and bit-flip phase errors (Y).

In the general case, possible errors are expressed as linear combinations of unitary error operators Ei

*Steane, A. (1998) –preprint – http://xxx.lanl.gov/abs/quant-ph/9708022

Given – an error correcting code,C, with a syndrome extraction operatorSC–

An n-bit quantum state |> is encoded in an n + k bit quantum state |>.

|> = C|>

Assume decoherence leads to an error state i e i E i| >

The signal encoded state|> can be recovered as follows:

- Apply the syndrome extraction operator SC to the quantum state (padded with sufficient 0 bits.) SC(i ei Ei | >) |0> = i ei Ei | > |i>
- Measure the |i> component of the result. This yields some random value i0 and projects the state to Ei0 | , i0>
- Apply the inverse error Ei0-1 to the first n+ k qubits of Ei0| , i0> to get the corrected state | >

Example

Consider an error correcting code C that maps

|0> |000>, and

|1> |111>

C can correct single bit-flip errors

E = {I I I, X I I, I X I, I I X}

The syndrome extraction operator is:

S: |x0, x1, x2, 0, 0, 0> |x0, x1, x2, x0xor x1, x0xor x2, x1xor x2>

Consider the qubit

|> = 1/2(|0> - |1>)

C|> = |> = 1/2(|000> - |111>)

And the error is (two single bit-flip errors)

E =(4/5) X I I + (3/5) I X I

The resulting error state is

E |> = (4/5 X I I + 3/5 I X I )(1/2) (|000> - |111>)

continuing

E |> = (4/52)|(100> - |011>) + (3/52)(|010> - |101>)

Now apply the syndrome extraction operator to

Sc E |> |000>

= Sc((4/52)|(100000> - |011000>) + (3/52)(|010000> - |101000>))

= (4/52)|(100110> - |011110>) + (3/52)(|010101> - |101101>)

= (4/52)|(100> - |011>) |110> + (3/52)(|010> - |101>) |101>

Measurement of the last three bits yields either state

|110> or |101>

Assume measurement yields |110>

then the state collapses to

1/2(|100> - |011>) |110>

The error can now be removed by applying Ei-1 = X I I to the first three bits of the state.

Ei-1 (1/2(|100> - |011>) = 1/2(|000> - |111>

The Architecture of a Quantum Computer

Preliminaries

Define an operator q † = |1><0|

q†|0> = |1><0| 0> = |1>

q†|1> = |1><0| 1> = 0

Similarly, q, the complex conjugate of q†, is defined

q = |0><1|

q|0> = |0><1| 0> = 0

q|1> = |0><1| 1> = |0>

Note that

q† + q = X and q†q + q q† = I

The Architecture of Quantum Computers

Time evolution of a quantum system

Let |in> be the initial state of a quantum system, then

the time evolution of this system is given by the equation

|out> = eiHt|in>

where H is the time evolution operator (Hamiltonian) and eiHt can be

expanded as

eiHt = 1 + iHt – H2t2/2! + …..

The operator H is operating an innumerable number of times, and our output state is the superposition of all these possibilities

.

.

.

1

0

logic

program

Program counter

Memory

The Architecture for a Quantum Computer

The basic Computer Architecture

k

k-1

.

.

.

|1>

|0>

|0>

Ak-1 Ak-2 ,.,A2 A1

|1>

A2

|1>

2

1

0

|0>

A1

|1>

|0>

PC

Qubit registers

q† 2q1 A2

The Architecture of a Quantum Computer

Are we there yet?

A1|>

A2A1 |>

q† 1q0 A1

H = q† i+1q i Ai+1 + c.c where summation is from i = 0 to i = k-1

Los Alamos preprint server: http://xxx.lanl.gov/archive/quant-ph

E. Reiffel and W. Polak. 1998. An Introduction to Quantum Computing for Non-Physicists. lanl preprint.

R. P. Feynman. 1996. Lectures on Computation. Addison-Wesley.

P. W. Schor. 1997. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. Society for Industrial and Applied Mathematics Journal on Computing 25, 5, 1484-1509. (Expanded version of 1994 paper)

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